Mathematics · Geometry
Triangle Calculator — Solve Any Triangle (SSS, SAS, ASA, AAS, SSA)
Give any three parts of a triangle (at least one a side) and get the rest: every side and angle, the area by Heron's formula, perimeter, the three heights, inradius and circumradius, and whether it's acute, right or obtuse. Handles SSS, SAS, ASA, AAS, the ambiguous SSA case (0, 1 or 2 triangles) and right triangles.
Your inputs
Quick start — try a typical case
In a right triangle, c is the hypotenuse.
Your result Live
Area
14.696938 cm²
Sides 5, 6, 7 cm · angles 44.4153°, 57.1217°, 78.463°
- Sides a, b, c
- 5 cm · 6 cm · 7 cm
- Angles A, B, C
- 44.4153° · 57.1217° · 78.463°
- Perimeter
- 18 cm
- Type
- scalene, acute
- Heights to a, b, c
- 5.8787754 · 4.8989795 · 4.1991253 cm
- Inradius · Circumradius
- 1.6329932 · 3.5721725 cm
How this is calculated
Law of cosines c² = a² + b² − 2ab·cos C · Law of sines a/sin A = b/sin B = c/sin C · Heron: area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2
- a, b, c
- Sides opposite angles A, B, C
Three parts fix a triangle (as long as one is a side). With three sides or two sides and the angle between them, the law of cosines gives the rest; with two angles, the third is 180° minus them and the law of sines gives the sides. Two sides and an angle that isn't between them can fit no triangle, one, or two different triangles — the 'ambiguous case' — and the calculator shows both.
Assumptions & sources
- Angles in degrees; a flat (Euclidean) triangle.
Calculator ID MATH-TRIANGLE-001 · version 1.0.0
Common questions
Can any three sides make a triangle?
Only if the two shorter sides add up to more than the longest (the triangle inequality). 3, 4 and 8 can't.
Why does SSA sometimes give two answers?
The side opposite the known angle can swing to touch the third side in two places, making one acute and one obtuse triangle.
How do I find the area from three sides?
Heron's formula: s = half the perimeter, area = √(s(s − a)(s − b)(s − c)). For 5, 6, 7: s = 9, area = √(9 × 4 × 3 × 2) ≈ 14.70.